COMPLETE METRICS CONFORMAL TO THE HYPERBOLIC DISC

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 97, Number 1, May 1986 COMPLETE METRICS CONFORMAL TO THE HYPERBOLIC DISC J. BLAND AND MORIS KALKA1 ABSTRACT. In this paper we study complete metrics conformai to the hyperbolic disc. We show that any smooth function K bounded between two negative constants is the curvature of such a metric. We also show that if K > 0 near the boundary, K cannot be the curvature of such a metric. Introduction. Here we consider complete metrics which are conformally equivalent to the hyperbolic metric on the unit disc in R2. If we denote the hyperbolic metric by h = dxdy/(l r2)2 and if g is conformally equivalent to h, then g is of the form e2u dx dy 9= (1-r2)2 for some function u. If we denote the Gauss curvature of g by Kg, then u satisfies the differential equation (*) Ahu = -1-Kge2u, where Ah is the hyperbolic Laplacian which henceforth we denote by Ä. We pose the question: For which smooth functions K can equation (*) be solved so that the resulting metric is complete? We also discuss the uniqueness of the solution for a given K. We can summarize our results as follows: Existence. Suppose there exist complete metrics gx, gi conformai to h with f/i < 92 and Kgi < K92. Let K9l < K < Kg2 be given. Then there exists a complete metric g with </i < </ < </2 and with Gauss curvature equal to K. In particular, if c^ < K < ci < 0, then there is a complete metric uniformly and conformally equivalent to h with curvature equal to K. In this case, the metric will also be unique. We will exhibit examples of complete metrics with curvature tending to -co at the boundary, complete metrics with curvature tending to 0 at the boundary, and metrics which have some positive curvature. Nonexistence. There are no complete metrics conformally equivalent to h with curvature nonnegative in a neighborhood of the boundary. Uniqueness. If K is bounded between two negative constants, then the solution to (*) is unique. However, there exist functions K which tend to a negative constant Received by the editors January 22, 1985 and, in revised form, May 24, Mathematics Subject Classification. Primary 53C99; Secondary 35J70. 1 Both authors acknowledge support from the Committee on Research of Tulane University American Mathematical Society /86 $ $.25 per page

2 COMPLETE METRICS CONFORMAL TO THE HYPERBOLIC DISC 129 near the boundary and for which there are distinct uniformly equivalent conformai metrics with curvature equal to if. Boundedness. If K < c < 0 in a neighborhood of the boundary, then the solution is bounded from above, while if K > N in a neighborhood of the boundary the solution is bounded below. These follow readily from the local Schwarz lemma [2]. We will also show that if K > 0 near the boundary, there is no complete metric uniformly and conformally equivalent to h with curvature equal to if. The same result holds if if >-co near the boundary. Similar results on the existence and uniqueness of solutions have been independently obtained by P. Aviles and R. McOwen [1]; they have also extended their results to the case n > Existence. THEOREM 1. Suppose tp, (f> and K are smooth functions on B = {x G R2 x < 1} with the properties that (i) <p < i in B, and (ii) A0 > -1 - if(x)e2^, Atp < -1 - if(x)e2^. Then there exists u G C (B) with (f> < u < ip such that Au = 1 K(x)e2u. PROOF. Fix a smooth function <p < g < ip on B. For R < 1, set Br = {x G R2 x < R}. On Br the operator A is uniformly elliptic. Thus by standard elliptic theory (see, for example, [3]) we can find ur G C (Br) satisfying Aur = -1 - if (x)e2ur in BR and <t><ur<ip, UR\dBR = g\dbr- We show that a subsequence of the ur converges uniformly on compact sets to a smooth solution u. It is clear that <f> < ur < if on Br> for all R > R'. We now proceed to get uniform gradient estimates on Br> for all u r, R > R" R' + ^(1 R'). It is well known that the inequality Mx,asR Vuñ \o,br < C(\ur\qiBr + \d2xtdballaeur\o,br,,) holds, where C is a constant depending only on dimension, Aß is the Laplacian with respect to the Euclidean (flat) metric, Vu is the norm with respect to the Euclidean metric and dx^sr denotes the distance from x to the boundary of Br. Applying this to ur for x G Bri, one obtains 5(1 - R')\Vur\0,br, < \dx,dbr Vufl \o,brl, < C(\ur\o,Br + \dl,dbr,aeur\o,br ) < C(\ur\0,Br + \d2xdb^eur\o,br,,)- We note that d2 9BAeur is uniformly equivalent to Aur. Since ur satisfies (*) in Br, the right-hand side of the inequality is bounded by C(\ur\o,br + I if(x)e2uñ 0,Bñ )

3 130 J. BLAND AND MORRIS KALKA which in turn is bounded independently of R. Thus we can let R 1 and we get a uniform C^-bound on ur in uri for R > R' + (1 - R'). Standard elliptic theory now gives us uniform estimates on the C2+a norm of ur independent of R. Let i?y be an increasing sequence of numbers 0 < Rj < Rj+i < 1 tending to 1. On Brx the sequence {urj} =2 1S uniformly bounded in 2+q,bRi and hence has a subsequence (tti,y) which converges to a solution. On Br2, the sequence Migare uniformly bounded in the \2+a,BR norm (for j sufficiently large). Hence, we extract a subsequence U2,j which converges uniformly on Br2. Continue inductively in this fashion. Then a subsequence of the diagonal sequence will converge uniformly on compact subsets to a solution. COROLLARY. Suppose that cx < K < C2 < 0. function u which solves Au = 1 K(x)e2u. Then there exists a smooth We now give examples of metrics e2uh which exhibit, in turn, the following properties: (i) if > 0 at some points, (ii) if -co as r -> 1, (iii) if -> 0 as r -> 1. We will see in 2 that any metric which exhibits property (ii) or (iii) cannot be uniformly equivalent to h. Thus the solutions cannot be bounded. (i), (iii). Let <f> (1 - r2)p. Then one computes that Ac/) = i>(4p2r2-4p). If g = e2*h, then Kg = -(A<p + l)e"2* = -(1 + <A(4p2r2-4p))e"2(1-r2)P. If p = 1, then g is a complete metric with positive curvature at 0. If p = 1, then g = e2lx~r h is a complete metric and ifg = -(1 + 0(4r2 + 4))e~2/1_r tends to 0 at r = 1. (ii) Let (p G C (B) have the property that <p = C ^ 0 near r = 0 and <f> = ln( ln(l r2)) near r = 1. Then near r 1, and if g = e^h, A4> = 4[r2(ln(l - r2))"2 + (ln(l - r2))"1] Kg = -[1 + 4{r2(ln(l - r2))"2 + (ln(l - r2)yl}\(\n(l which tends to oo. We see that g is complete since for u = (1 r2). 2. Nonexistence. gij = (1/ulnu)2^ - r2))2 THEOREM 2. Suppose K G C7 (ß) is nonnegative in a neighborhood of db. Then there does not exist any complete metric g, conformai to h, having curvature if. PROOF. Suppose o = e2uh is such a metric. Consider the identity mapping (B,g) >(B,h). By the local Schwarz lemma [2], the map is metric decreasing, up to a factor. Thus e2"/(l - r2)2 > C/(l - r2)2 and u is bounded below.

4 COMPLETE METRICS CONFORMAL TO THE HYPERBOLIC DISC 131 Thus we may assume that u > C satisfies Au = 1 K(x)e2u. We denote by ü(r) the average of u on circles of radius r. Since A is rotationally symmetric, Au = Aü. As if is nonnegative near the boundary, it follows that Aü < 1 near db. Thus (1-r2)2 d ( du\ <_ r dr \ dr We may integrate this inequality explicitly to see that ü > oo. Hence there is a sequence pi > db such that u(pi) > oo, contradicting the lower boundedness ofii. D PROPOSITION 3. If K > 0 or K -» -oo near db, there do not exist any conformai metrics uniformly equivalent to h, with curvature K. (That is, there do not exist any bounded solutions to (*).) PROOF. First suppose if 0 near db and g = e2uh has curvature if. Thus Au 1 near db. In particular Au < c < 0 near db. By radial averaging as in the proof of Theorem 2, we see that u is not bounded below. Now suppose that if oo near db and g = e2uh has curvature if. Since if = (Au + l)e_2u we see that Au +oo. Using the radial averaging technique again, we see that there is a sequence p with u(p ) > oo. Ü REMARK. It is an immediate consequence of the local Schwarz lemma that (i) if if < near db, then u is bounded above, and (ii) if if > N near db, then u is bounded below. 3. Uniqueness. PROPOSITION 4. Let cx < K < C2 < 0. Then there exists a unique complete metric g conformai to h, with curvature if. PROOF. Existence has already been shown. Thus let gi = e2uh and 02 = e2vh both have curvature if. Then A(u v) = K(e2u e2v). By the remark above, u and v are both bounded. Thus (u v) is bounded. By the generalized maximum principle [4], there is a sequence of points pk for which limfc_<00(u v)(pk) = infß(u - v) and limfc_oc,a(u - v) > 0. Thus lim/t oo(-if (e2u - e2")) > 0 implying limfe^00(u v) > 0. A similar argument shows that sup(u v) < 0. Thus u = v. THEOREM 5. There exists K with the following properties: (i) K tends to a negative constant near db. (ii) There are two distinct complete metrics gi e2uh and gi e2vh uniformly and conformally equivalent to h and both having curvature if. PROOF. Let <j> G C (B) satisfy (i) (j) jl 0 in B, (ii) <f> = 1 in a neighborhood of the origin, (iii) <j> = (1 - r2)l/2 + V3/2 = (! _ r2)a near r = L Define / = Ar/>/(e2* - 1). Then / is a smooth function which is identically 0 near the origin. We have seen in 1 that A(l - r2)a = (1 - r2)a(4a2r2-4a). Thus, near r = 1, / is approximately equal to (1 - r2)q(4a2r2-4a)/2(l - r2)a. By the choice of a, we see that / = 1 + 0(1 - r2).

5 132 J. BLAND AND MORRIS KALKA It is thus possible to find a bounded solution, v G C (B), of the equation Av = f - 1. We define if = -(Au + l)e~2v and u = v + (f>. Then Av = -1 - if e2v by definition and Au = (/ - 1) + /(e2* - 1) = (Au + l)e2ue~2v - 1 = -1 - ife2u. It is easy to check that if tends to a negative constant near db. D NOTE ADDED IN PROOF. Because of the normalization involved in our choice of hyperbolic metric h, equation (*) should read: A/ u = 4 Kge2u. This will not affect any of the results, but it will lead to corresponding changes throughout the paper. REFERENCES 1. P. Aviles and R. McOwen, Conformai deformations of complete manifolds with negative curvature, J. Differential Geom. 21 (1985), J. Bland, Local boundary regularity of the canonical Köhler-Einstein metric on pseudoconvex domains, Math. Ann. 263 (1983), R. Courant and D. Hubert, Methods of mathematical physics. II, Interscience, New York, S. T. Yau, Harmonic unctions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), DEPARTMENT OF MATHEMATICS, TULANE UNIVERSITY, NEW ORLEANS, LOUISIANA 70118